Question: The sum of two numbers is $19$ and their difference is $5$. What is their product?
Solution: Let the numbers be $x$ and $y$. The problem translates into a system of equations: \begin{align*}
x+y &= 19\\
x-y &= 5.
\end{align*} Adding the equations, we obtain $x+y+x-y = 24$, which implies $2x = 24$. Therefore, $x = 12$. Subtracting the equations, we obtain $(x+y)-(x-y) = 14$, which implies $2y = 14$. Therefore, $y = 7$. Our desired product is $xy = 12\cdot7 =\boxed{84}$.